In a region of space, both electric and magnetic field are present simultaneously in opposite direction. A positively charged particle is projected with certain speed an angle `theta (lt 90^(@))` with magnetic field. It will move in a

Options

A. Helical path of uniform pitch
B. Helical path of increasing pitch
C. Helical path of decreasing pitch
D. Helical path, whose pitch first decreases and then increases

Answer

To solve the problem, we need to analyze the motion of a positively charged particle projected in a region where both electric and magnetic fields are present, and they are directed oppositely. The particle is projected at an angle \( \theta \) with respect to the magnetic field.

### Step-by-Step Solution:

1. **Understanding the Forces Acting on the Particle:**
- The positively charged particle experiences a magnetic force due to its motion in the magnetic field and an electric force due to the electric field.
- The magnetic force \( \mathbf{F}_B \) is given by \( \mathbf{F}_B = q(\mathbf{v} \times \mathbf{B}) \), where \( q \) is the charge, \( \mathbf{v} \) is the velocity of the particle, and \( \mathbf{B} \) is the magnetic field.
- The electric force \( \mathbf{F}_E \) is given by \( \mathbf{F}_E = q\mathbf{E} \), where \( \mathbf{E} \) is the electric field.

2. **Decomposing the Velocity:**
- The velocity \( \mathbf{v} \) of the particle can be decomposed into two components:
- \( v \cos \theta \): the component along the magnetic field.
- \( v \sin \theta \): the component perpendicular to the magnetic field.

3. **Motion in the Magnetic Field:**
- The component \( v \sin \theta \) causes the particle to move in a circular path due to the magnetic force, which acts perpendicular to the velocity.
- The radius of this circular motion is determined by the magnetic force and the centripetal force.

4. **Effect of the Electric Field:**
- The electric field is in the opposite direction to the component \( v \cos \theta \).
- This means that the electric field will exert a force that opposes the motion of the particle along the direction of \( v \cos \theta \), causing a retardation effect.

5. **Analyzing the Path:**
- Initially, as the particle moves, the pitch of the helical path (the distance between successive turns of the helix) will decrease because the component of velocity along the magnetic field is decreasing due to the electric field's opposing force.
- Eventually, the particle will stop moving in the positive direction and start moving back towards the negative direction due to the electric field.

6. **Change in Pitch:**
- After the particle reverses direction, it will accelerate in the opposite direction due to the electric field, causing the pitch of the helical path to increase.
- Thus, the motion of the particle is characterized by a helical path whose pitch first decreases and then increases.

### Conclusion:
The correct answer is that the particle moves in a helical path whose pitch first decreases and then increases.